Trade-Offs Between Size and Degree in Polynomial Calculus

Building on [Clegg et al. \u2796], [Impagliazzo et al. \u2799] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(?(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen \u2716], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary

Extremely Deep Proofs

We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c ? n^{1-?}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-?}/c must necessarily have depth ? n^c. By setting c = o(n^{1-?}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems

Realising superoscillations: A review of mathematical tools and their application

Superoscillations are making a growing impact on an ever-increasing number of real-world applications, as early theoretical analysis has evolved into wide experimental realisation. This is particularly true in optics: the first application area to have extensively embraced superoscillations, with much recent growth. This review provides a tool for anyone planning to expand the boundaries in an application where superoscillations have already been used, or to apply superoscillations to a new application. By reviewing the mathematical methods for constructing superoscillations, including their considerations and capabilities, we lay out the options for anyone wanting to construct a device that uses superoscillations. Superoscillations have inherent trade-offs: as the size of spot reduces, its relative intensity decreases as high-energy sidebands appear. Different methods provide solutions for optimising different aspects of these trade-offs, to suit different purposes. Despite numerous technological ways of realising superoscillations, the mathematical methods can be categorised into three approaches: direct design of superoscillatory functions, design of pupil filters and design of superoscillatory lenses. This categorisation, based on mathematical methods, is used to highlight the transferability of methods between applications. It also highlights areas for future theoretical development to enable the scientific and technological boundaries to be pushed even further in real-world applications

Modeling and model-aware signal processing methods for enhancement of optical systems

Theoretical and numerical modeling of optical systems are increasingly being utilized in a wide range of areas in physics and engineering for characterizing and improving existing systems or developing new methods. This dissertation focuses on determining and improving the performance of imaging and non-imaging optical systems through modeling and developing model-aware enhancement methods. We evaluate the performance, demonstrate enhancements in terms of resolution and light collection efficiency, and improve the capabilities of the systems through changes to the system design and through post-processing techniques. We consider application areas in integrated circuit (IC) imaging for fault analysis and malicious circuitry detection, and free-form lens design for creating prescribed illumination patterns. The first part of this dissertation focuses on sub-surface imaging of ICs for fault analysis using a solid immersion lens (SIL) microscope. We first derive the Green's function of the microscope and use it to determine its resolution limits for bulk silicon and silicon-on-insulator (SOI) chips. We then propose an optimization framework for designing super-resolving apodization masks that utilizes the developed model and demonstrate the trade-offs in designing such masks. Finally, we derive the full electromagnetic model of the SIL microscope that models the image of an arbitrary sub-surface structure. With the rapidly shrinking dimensions of ICs, we are increasingly limited in resolving the features and identifying potential modifications despite the resolution improvements provided by the state-of-the-art microscopy techniques and enhancement methods described here. In the second part of this dissertation, we shift our focus away from improving the resolution and consider an optical framework that does not require high resolution imaging for detecting malicious circuitry. We develop a classification-based high-throughput gate identification method that utilizes the physical model of the optical system. We then propose a lower-throughput system to increase the detection accuracy, based on higher resolution imaging to supplement the former method. Finally, we consider the problem of free-form lens design for forming prescribed illumination patterns as a non-imaging application. Common methods that design free-form lenses for forming patterns consider the input light source to be a point source, however using extended light sources with such lenses lead to significant blurring in the resulting pattern. We propose a deconvolution-based framework that utilizes the lens geometry to model the blurring effects and eliminates this degradation, resulting in sharper patterns

Modeling and model-aware signal processing methods for enhancement of optical systems

Theoretical and numerical modeling of optical systems are increasingly being utilized in a wide range of areas in physics and engineering for characterizing and improving existing systems or developing new methods. This dissertation focuses on determining and improving the performance of imaging and non-imaging optical systems through modeling and developing model-aware enhancement methods. We evaluate the performance, demonstrate enhancements in terms of resolution and light collection efficiency, and improve the capabilities of the systems through changes to the system design and through post-processing techniques. We consider application areas in integrated circuit (IC) imaging for fault analysis and malicious circuitry detection, and free-form lens design for creating prescribed illumination patterns. The first part of this dissertation focuses on sub-surface imaging of ICs for fault analysis using a solid immersion lens (SIL) microscope. We first derive the Green's function of the microscope and use it to determine its resolution limits for bulk silicon and silicon-on-insulator (SOI) chips. We then propose an optimization framework for designing super-resolving apodization masks that utilizes the developed model and demonstrate the trade-offs in designing such masks. Finally, we derive the full electromagnetic model of the SIL microscope that models the image of an arbitrary sub-surface structure. With the rapidly shrinking dimensions of ICs, we are increasingly limited in resolving the features and identifying potential modifications despite the resolution improvements provided by the state-of-the-art microscopy techniques and enhancement methods described here. In the second part of this dissertation, we shift our focus away from improving the resolution and consider an optical framework that does not require high resolution imaging for detecting malicious circuitry. We develop a classification-based high-throughput gate identification method that utilizes the physical model of the optical system. We then propose a lower-throughput system to increase the detection accuracy, based on higher resolution imaging to supplement the former method. Finally, we consider the problem of free-form lens design for forming prescribed illumination patterns as a non-imaging application. Common methods that design free-form lenses for forming patterns consider the input light source to be a point source, however using extended light sources with such lenses lead to significant blurring in the resulting pattern. We propose a deconvolution-based framework that utilizes the lens geometry to model the blurring effects and eliminates this degradation, resulting in sharper patterns

On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas

Microgravity Combustion Diagnostics Workshop

Through the Microgravity Science and Applications Division (MSAD) of the Office of Space Science and Applications (OSSA) at NASA Headquarters, a program entitled, Advanced Technology Development (ATD) was promulgated with the objective of providing advanced technologies that will enable the development of future microgravity science and applications experimental flight hardware. Among the ATD projects one, Microgravity Combustion Diagnostics (MCD), has the objective of developing advanced diagnostic techniques and technologies to provide nonperturbing measurements of combustion characteristics and parameters that will enhance the scientific integrity and quality of microgravity combustion experiments. As part of the approach to this project, a workshop was held on July 28 and 29, 1987, at the NASA Lewis Research Center. A small group of laser combustion diagnosticians met with a group of microgravity combustion experimenters to discuss the science requirements, the state-of-the-art of laser diagnostic technology, and plan the direction for near-, intermediate-, and long-term programs. This publication describes the proceedings of that workshop